Some relations between entropy and approximation numbers

A general result is obtained which relates the entropy numbers of compact maps on Hilbert space to its approximation numbers. Compared with previous works in this area, it is particularly convenient for dealing with the cases where the approximation numbers decay rapidly. A nice estimation between entropy and approximation numbers for noncompact maps is given. Partially supported by the National 863 Project of China and the National Natural Science Foundation of China (Grant No. 19771003).     

Some remarks on entropy numbers of sets with non-integer dimension

Abstract  We study entropy numbers of sets and generalize some results shown by B. Carl and I. Stephani in [2]. It is well known that if Ω is the closed unit ball in the Euclidean space , then for every ,

Asymptotic estimates for the approximation and entropy numbers of a one-weight Riemann-Liouville operator

We obtain two-sided estimates for the asymptotic behavior of the approximation and entropy numbers of a one-weight Riemann-Liouville operator of an arbitrary integer order acting in Lebesgue spaces on the semiaxis.     

Some remarks on matrix norms,condition numbers,and error estimates for linear equations

Inequalities between some norms of rectangular matrices and the corresponding relationships between condition numbers are established and clearly arranged in two simple diagrams. Furthermore, some well-known error estimates for linear equations in the nonsingular case are shown to be valid for all submultiplicative matrix norms. The new proofs work without using infinite series.     

Pseudorandom numbers and entropy conditions

We investigate measures of pseudorandomness of finite sequences (x_n) of real numbers. Mauduit and Sárközy introduced the “well-distribution measure”, depending on the behavior of the sequence (x_n) along arithmetic subsequences (x_ak+b). We extend this definition by replacing the class of arithmetic progressions by an arbitrary class of sequences of positive integers and show that the so obtained measure is closely related to the metric entropy of the class . Using standard probabilistic techniques, this fact enables us to give precise bounds for the pseudorandomness measure of classical constructions. In particular, we will be interested in “truly” random sequences and sequences of the form {n_kω}, where {·} denotes fractional part, (n_k) is a given sequence of integers and ω[0,1).     

More on the duality conjecture for entropy numbers

We verify, up to a logarithmic factor, the duality conjecture for entropy numbers in the case where one of the bodies is an ellipsoid. To cite this article: S. Artstein et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Low regularity classes and entropy numbers

We note a sharp embedding of the Besov space into exponential classes and prove entropy estimates for the compact embedding of subclasses with logarithmic smoothness, considered by Kashin and Temlyakov. A.S. was supported in part by the National Science Foundation grant 0652890.     

Quantum lower bounds by entropy numbers

We use entropy numbers in combination with the polynomial method to derive a new general lower bound for the nth minimal error in the quantum setting of information-based complexity. As an application, we improve some lower bounds on quantum approximation of embeddings between finite dimensional L_p spaces and of Sobolev embeddings.     

Some estimates for large deviations and their application to strong law of large numbers

Siberian Mathematical Journal -     

Some remarks on conditional entropy

Summary Using a definition of conditional entropy given by Hanen and Neveu [5, 10, 11] we discuss in this paper some properties of conditional entropy and mean entropy, in particular an integral representation of conditional entropy (§ 2), and the decomposition theorem of the KolmogorovSina¯i invariant (§ 3) (see also [6–8] and [12]).There is an essential difference between Jacob's proof of the last called theorem [6] and the proof given below. The definition of a Lebesgue space, given by Rokhlin [7–9] and [12], is not used in this paper.     

Amenable groups, topological entropy and Betti numbers

We investigate an analogue of theL ²-Betti numbers for amenable linear subshifts. The role of the von Neumann dimension shall be played by the topological entropy. Partially supported by OTKA grant T 25004 and the Bolyai Fellowship.     

Improved entropy decay estimates for the heat equation

Improved entropy decay estimates for the heat equation are obtained by selecting well-parametrized Gaussians. Either by mass centering or by fixing the second moments or the covariance matrix of the solution, relative entropy toward these Gaussians is shown to decay with better constants than classical estimates.     

Some connected ramsey numbers

A graph G is co-connected if both G and its complement ? are connected and nontrivial. For two graphs A and B, the connected Ramsey number r_c(A, B) is the smallest integer n such that there exists a co-connected graph of order n, and if G is a co-connected graph on at least n vertices, then A ? G or B ? ?. If neither A or B contains a bridge, then it is known that r_c(A, B) = r(A, B), where r(A, B) denotes the usual Ramsey number of A and B. In this paper r_c(A, B) is calculated for some pairs (A, B) when r(A, B) is known and at least one of the graphs A or B has a bridge. In particular, r_c(A, B) is calculated for A a path and B either a cycle, star, or complete graph, and for A a star and B a complete graph.     

Some remarks concerning kissing numbers,blocking numbers and covering numbers

This article shows an inequality concerning blocking numbers and Hadwiger's covering numbers and presents a strange phenomenon concerning kissing numbers and blocking numbers. As a simple corollary, we can improve the known upper bounds for Hadwiger's covering numbers ford-dimensional centrally symmetric convex bodies to 3 ^d –1.     

Some small ramsey numbers

In previous work, the Ramsey numbers have been evaluated for all pairs of graphs with at most four points. In the present note, Ramsey numbers are tabulated for pairs F₁, F₂ of graphs where F₁ has at most four points and F₂ has exactly five points. Exact results are listed for almost all of these pairs.     

Some estimates in spaces

We consider a class of rearrangement invariant Banach Function Spaces recently appeared in a paper by the same authors, containing at the same time some Lorentz spaces Γ(ν), classical Lebesgue spaces and small Lebesgue spaces. We discuss the main properties coming directly from the norm, and, for certain values of the involved parameters, we prove some estimates of the norm of the associate space.